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  "body": "\n## Statement\n\nLet $f \\colon [-1, 1] \\to \\R$ be continuous and let $E_n(f)$ be the uniform distance on $[-1, 1]$ from $f$ to the polynomials of degree at most $n$.\nFor $\\rho \u003e 1$, let $\\mathcal{E}_\\rho$ be the open region bounded by the [[ellipse|0VKCCEXBJ1]] with foci $\\pm 1$ whose semi-axes have sum $\\rho$.\nThen $f$ extends to a [[holomorphic function|0NAHXR1FRH]] on $\\mathcal{E}_\\rho$ if and only if\n$$\\limsup_{n \\to \\infty} E_n(f)^{1/n} \\le \\frac{1}{\\rho}.$$\n\n## Remarks\n\nThe region $\\mathcal{E}_\\rho$ is the image of the annulus $1 \u003c \\abs{z} \u003c \\rho$ under the map $z \\mapsto (z + z^{-1})/2$, together with $[-1, 1]$.\nProof idea for the direct half: for $1 \u003c s \u003c \\rho$, the coefficients of the Chebyshev series of $f$ are $O(s^{-n})$, by a contour integral over the image of $\\abs{z} = s$.\nTruncating the series gives $E_n(f) = O(s^{-n})$.\nFor the converse, write $f = p_0 + \\sum (p_{n+1} - p_n)$ with best approximations $p_n$ and bound each difference on $\\mathcal{E}_s$ for $s \u003c \\rho$ by its size on $[-1, 1]$.\n\nExample: for $a \u003e 1$ and $f(x) = 1/(x - a)$, the pole $a$ lies on the boundary of $\\mathcal{E}_\\rho$ with $\\rho = a + \\sqrt{a^2 - 1}$.\nSo $E_n(f)$ decays like $(a + \\sqrt{a^2 - 1})^{-n}$, up to factors that grow slower than any exponential.\nFor the Runge function $1/(1 + 25x^2)$, the poles $\\pm i/5$ give $\\rho = (1 + \\sqrt{26})/5 \\approx 1.22$.\n\n## Explanations\n\n### L1 human\n\nFor an analytic function on $[-1, 1]$, the error of the best polynomial approximation shrinks geometrically with the degree.\nThe rate is set by how far the function extends into the complex plane, measured by the largest ellipse with foci $\\pm 1$ on which it stays analytic.\n\n### L2 human\n\nThis explains why Chebyshev methods converge so fast for analytic functions, and why nearby complex singularities slow them down even when $f$ looks harmless on the real line.\nIt is an inverse and a direct theorem at once: geometric decay of $E_n(f)$ and analyticity in an ellipse are equivalent.\nThe ellipses $\\partial \\mathcal{E}_\\rho$ are the level curves of the Green function of the complement of $[-1, 1]$, which leads to the general theory of Walsh for other sets.\n",
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